International Association
for Cryptologic Research

Fully homomorphic encryption (FHE) is an advanced cryptography technique to allow computations (i.e., addition and multiplication) over encrypted data. After years of effort, the performance of FHE has been significantly improved and it has moved from theory to practice. The transciphering framework is another important technique in FHE to address the issue of ciphertext expansion and reduce the client-side computational overhead. To apply the transciphering framework to the CKKS FHE scheme, a new transciphering framework called the Real-to-Finite-Field (RtF) framework and a corresponding FHE-friendly symmetric-key primitive called HERA were proposed at ASIACRYPT 2021. Although HERA has a very similar structure to AES, it is considerably different in the following aspects: 1) the power map x → x3 is used as the S-box; 2) a randomized key schedule is used; 3) it is over a prime field Fp with p > 216. In this work, we perform the first third-party cryptanalysis of HERA, by showing how to mount new algebraic attacks with multiple collisions in the round keys. Specifically, according to the special way to randomize the round keys in HERA, we find it possible to peel off the last nonlinear layer by using collisions in the last-round key and a simple property of the power map. In this way, we could construct an overdefined system of equations of a much lower degree in the key, and efficiently solve the system via the linearization technique. As a esult, for HERA with 192 and 256 bits of security, respectively, we could break some parameters under the same assumption made by designers that the algebra constant ω for Gaussian elimination is ω = 2, i.e., Gaussian elimination on an n × n matrix takes O(nω) field operations. If using more conservative choices like ω ∈ {2.8, 3}, our attacks can also successfully reduce the security margins of some variants of HERA to only 1 round. However, the security of HERA with 80 and 128 bits of security is not affected by our attacks due to the high cost to find multiple collisions. In any case, our attacks reveal a weakness of HERA caused by the randomized key schedule and its small state size.

The hash function RIPEMD-160 is an ISO/IEC standard and is being used to generate the bitcoin address together with SHA-256. Despite the fact that many hash functions in the MD-SHA hash family have been broken, RIPEMD-160 remains secure and the best collision attack could only reach up to 34 out of 80 rounds, which was published at CRYPTO 2019. In this paper, we propose a new collision attack on RIPEMD-160 that can reach up to 36 rounds with time complexity $2^{64.5}$. This new attack is facilitated by a new strategy to choose the message differences and new techniques to simultaneously handle the differential conditions on both branches. Moreover, different from all the previous work on RIPEMD-160, we utilize a MILP-based method to search for differential characteristics, where we construct a model to accurately describe the signed difference transitions through its round function. As far as we know, this is the first model targeting the signed difference transitions for the MD-SHA hash family. Indeed, we are more motivated to design this model by the fact that many automatic tools to search for such differential characteristics are not publicly available and implementing them from scratch is too time-consuming and difficult. Hence, we expect that this can be an alternative easy tool for future research, which only requires to write down some simple linear inequalities.

In this paper, we present 4 major contributions to ARX ciphers and in particular, to the Salsa/ChaCha family of stream ciphers: (a) We propose an improved differential-linear distinguisher against ChaCha. To do so, we propose a new way to approach the derivation of linear approximations by viewing the algorithm in terms of simpler subrounds. Using this idea, we show that it is possible to derive almost all linear approximations from previous works from just 3 simple rules. Furthermore, we show that with one extra rule, it is possible to improve the linear approximations proposed by Coutinho and Souza at Eurocrypt 2021 (Coutinho and Neto, in: Canteaut, Standaert (eds) Advances in cryptology—EUROCRYPT 2021—40th annual international conference on the theory and applications of cryptographic techniques, Zagreb, Croatia, October 17–21, 2021, proceedings, Part I. Lecture notes in computer science, vol 12696, Springer, 2021). (b) We propose a technique called Bidirectional Linear Expansions (BLE) to improve attacks against Salsa. While previous works only considered linear expansions moving forward into the rounds, BLE explores the expansion of a single bit in both forward and backward directions. Applying BLE, we propose the first differential-linear distinguishers reaching 7 and 8 rounds of Salsa and we improve Probabilistic Neutral Bit (PNB) key-recovery attacks against 8 rounds of Salsa. (c) At Eurocrypt 2022 (Dey et al in Revamped differential-linear cryptanalysis on reduced round chacha, Springer, 2022), Dey et al. proposed a technique to combine two input–output positions in a PNB attack. In this paper, we generalize this technique for an arbitrary number of input–output positions. Combining this approach with BLE, we are able to improve key recovery attacks against 7 rounds of Salsa. (d) Using all the knowledge acquired studying the cryptanalysis of these ciphers, we propose some modifications in order to provide better diffusion per round and higher resistance to cryptanalysis, leading to a new stream cipher named Forró. We show that Forró has higher security margin; this allows us to reduce the total number of rounds while maintaining the security level, thus creating a faster cipher in many platforms, especially in constrained devices. (e) Finally, we developed CryptDances , a new tool for the cryptanalysis of Salsa, ChaCha, and Forró designed to be used in high performance environments with several GPUs. With CryptDances it is possible to compute differential correlations, to derive new linear approximations for ChaCha automatically, to automate the computation of the complexity of PNB attacks, among other features. We make CryptDances available for the community at https://github.com/murcoutinho/cryptDances .

The LWE problem is one of the prime candidates for building the most efficient post-quantum secure public key cryptosystems. Many of those schemes, like Kyber, Dilithium or those belonging to the NTRU-family, such as NTRU-HPS, -HRSS, BLISS or GLP, make use of small max norm keys to enhance efficiency. The best attack on these schemes is a hybrid attack, which combines combinatorial techniques and lattice reduction. While lattice reduction is not known to be able to exploit the small max norm choices, May recently showed (Crypto 2021) that such choices allow for more efficient combinatorial attacks. However, these combinatorial attacks suffer enormous memory requirements, which render them inefficient in realistic attack scenarios and, hence, make their general consideration when assessing security questionable. Therefore, more memory-efficient substitutes for these algorithms are needed. In this work, we provide new combinatorial algorithms for recovering small max norm LWE secrets using only a polynomial amount of memory. We provide analyses of our algorithms for secret key distributions of current NTRU, Kyber and Dilithium variants, showing that our new approach outperforms previous memory-efficient algorithms. For instance, considering uniformly random ternary secrets of length $n$ we improve the best known time complexity for polynomial memory algorithms from $2^{1.063n}$ down-to $2^{0.926n}$. We obtain even larger gains for LWE secrets in $\{-m,\ldots,m\}^n$ with $m=2,3$ as found in Kyber and Dilithium. For example, for uniformly random keys in $\{-2,\ldots,2\}^n$ as is the case for Dilithium we improve the previously best time from $2^{1.742n}$ down-to $2^{1.282n}$. Our fastest algorithm incorporates various different algorithmic techniques, but at its heart lies a nested collision search procedure inspired by the Nested-Rho technique from Dinur, Dunkelman, Keller and Shamir (Crypto 2016). Additionally, we heavily exploit the representation technique originally introduced in the subset sum context to make our nested approach efficient.

At Asiacrypt 2017, Rønjom et al. presented key-independent distinguishers for different numbers of rounds of AES, ranging from 3 to 6 rounds, in their work titled “Yoyo Tricks with AES”. The reported data complexities for these distinguishers were 3, 4, 225.8, and 2122.83, respectively. In this work, we revisit those key-independent distinguishers and analyze their success probabilities.We show that the distinguishing algorithms provided for 5 and 6 rounds of AES in the paper of Rønjom et al. are ineffective with the proposed data complexities. Our thorough theoretical analysis has revealed that the success probability of these distinguishers for both 5-round and 6-round AES is approximately 0.5, with the corresponding data complexities mentioned earlier.We investigate the reasons behind this seemingly random behavior of those reported distinguishers. Based on our theoretical findings, we have revised the distinguishing algorithm for 5-round AES. Our revised algorithm demonstrates success probabilities of approximately 0.55 and 0.81 for 5-round AES, with data complexities of 229.95 and 230.65, respectively. We have also conducted experimental tests to validate our theoretical findings, which further support our findings.Additionally, we have theoretically demonstrated that improving the success probability of the distinguisher for 6-round AES from 0.50000 to 0.50004 would require a data complexity of 2129.15. This finding invalidates the reported distinguisher by Rønjom et al. for 6-round AES.

We address Partial Key Exposure attacks on CRT-RSA on secret exponents $d_p, d_q$ with small public exponent $e$. For constant $e$ it is known that the knowledge of half of the bits of one of $d_p, d_q$ suffices to factor the RSA modulus $N$ by Coppersmith's famous {\em factoring with a hint} result. We extend this setting to non-constant $e$. Somewhat surprisingly, our attack shows that RSA with $e$ of size $N^{\frac 1 {12}}$ is most vulnerable to Partial Key Exposure, since in this case only a third of the bits of both $d_p, d_q$ suffices to factor $N$ in polynomial time, knowing either most significant bits (MSB) or least significant bits (LSB). Let $ed_p = 1 + k(p-1)$ and $ed_q = 1 + \ell(q-1)$. On the technical side, we find the factorization of $N$ in a novel two-step approach. In a first step we recover $k$ and $\ell$ in polynomial time, in the MSB case completely elementary and in the LSB case using Coppersmith's lattice-based method. We then obtain the prime factorization of $N$ by computing the root of a univariate polynomial modulo $kp$ for our known $k$. This can be seen as an extension of Howgrave-Graham's {\em approximate divisor} algorithm to the case of {\em approximate divisor multiples} for some known multiple $k$ of an unknown divisor $p$ of $N$. The point of {\em approximate divisor multiples} is that the unknown that is recoverable in polynomial time grows linearly with the size of the multiple $k$. Our resulting Partial Key Exposure attack with known MSBs is completely rigorous, whereas in the LSB case we rely on a standard Coppersmith-type heuristic. We experimentally verify our heuristic, thereby showing that in practice we reach our asymptotic bounds already using small lattice dimensions. Thus, our attack is highly practical.

In this paper, we provide several improvements over the existing differential-linear attacks on ChaCha. ChaCha is a stream cipher which has $20$ rounds. At CRYPTO $2020$, Beierle et al. observed a differential in the $3.5$-th round if the right pairs are chosen. They produced an improved attack using this, but showed that to achieve a right pair, we need $2^5$ iterations on average. In this direction, we provide a technique to find the right pairs with the help of listing. Also, we provide a strategical improvement in PNB construction, modification of complexity calculation and an alternative attack method using two input-output pairs. Using these, we improve the time complexity, reducing it to $2^{221.95}$ from $2^{230.86}$ reported by Beierle et al. for $256$ bit version of ChaCha. Also, after a decade, we improve existing complexity (Shi et al: ICISC 2012) for a $6$-round of $128$ bit version of ChaCha by more than 11 million times and produce the first-ever attack on 6.5-round ChaCha$128$ with time complexity $2^{123.04}.$

Side Channel Attack (SCA) exploits the physical information leakage (such as electromagnetic emanation) from a device that performs some cryptographic operation and poses a serious threat in the present IoT era. In the last couple of decades, there have been a large body of research works dedicated to streamlining/improving the attacks or suggesting novel countermeasures to thwart those attacks. However, a closer inspection reveals that a vast majority of published works in the context of symmetric key cryptography is dedicated to block ciphers (or similar designs). This leaves the problem for the stream ciphers wide open. There are few works here and there, but a generic and systematic framework appears to be missing from the literature. Motivating by this observation, we explore the problem of SCA on stream ciphers with extensive details. Loosely speaking, our work picks up from the recent TCHES’21 paper by Sim, Bhasin and Jap. We present a framework by extending the efficiency of their analysis, bringing it into more practical terms.In a nutshell, we develop an automated framework that works as a generic tool to perform SCA on any stream cipher or a similar structure. It combines multiple automated tools (such as, machine learning, mixed integer linear programming, satisfiability modulo theory) under one umbrella, and acts as an end-to-end solution (taking side channel traces and returning the secret key). Our framework efficiently handles noisy data and works even after the cipher reaches its pseudo-random state. We demonstrate its efficacy by taking electromagnetic traces from a 32-bit software platform and performing SCA on a high-profile stream cipher, TRIVIUM, which is also an ISO standard. We show pragmatic key recovery on TRIVIUM during its initialization and also after the cipher reaches its pseudo-random state (i.e., producing key-stream).

The security of the post-quantum signature scheme Picnic is highly related to the difficulty of recovering the secret key of LowMC from a single plaintext-ciphertext pair. Since Picnic is one of the alternate third-round candidates in NIST post-quantum cryptography standardization process, it has become urgent and important to evaluate the security of LowMC in the Picnic setting. The best attacks on LowMC with full S-box layers used in Picnic3 were achieved with Dinur’s algorithm. For LowMC with partial nonlinear layers, e.g. 10 S-boxes per round adopted in Picnic2, the best attacks on LowMC were published by Banik et al. with the meet-in-the-middle (MITM) method.In this paper, we improve the attacks on LowMC in a model where memory consumption is costly. First, a new attack on 3-round LowMC with full S-box layers with negligible memory complexity is found, which can outperform Bouillaguet et al.’s fast exhaustive search attack and can achieve better time-memory tradeoffs than Dinur’s algorithm. Second, we extend the 3-round attack to 4 rounds to significantly reduce the memory complexity of Dinur’s algorithm at the sacrifice of a small factor of time complexity. For LowMC instances with 1 S-box per round, our attacks are shown to be much faster than the MITM attacks. For LowMC instances with 10 S-boxes per round, we can reduce the memory complexity from 32GB (238 bits) to only 256KB (221 bits) using our new algebraic attacks rather than the MITM attacks, while the time complexity of our attacks is about 23.2 ∼ 25 times higher than that of the MITM attacks. A notable feature of our new attacks (apart from the 4-round attack) is their simplicity. Specifically, only some basic linear algebra is required to understand them and they can be easily implemented.

ZUC-256 is a stream cipher designed for 5G applications by the ZUC team. Together with AES-256 and SNOW-V, it is currently being under evaluation for standardized algorithms in 5G mobile telecommunications by Security Algorithms Group of Experts (SAGE). A notable feature of the round update function of ZUC-256 is that many operations are defined over different fields, which significantly increases the difficulty to analyze the algorithm.As a main contribution, with the tools of the modular difference, signed difference and XOR difference, we develop new techniques to carefully control the interactions between these operations defined over different fields. At first glance, our techniques are somewhat similar to those developed by Wang et al. for the MD-SHA hash family. However, as ZUC-256 is quite different from the MD-SHA hash family and its round function is much more complex, we are indeed dealing with different problems and overcoming new obstacles.As main results, by utilizing complex input differences, we can present the first distinguishing attacks on 31 out of 33 rounds of ZUC-256 and 30 out of 33 rounds of the new version of ZUC-256 called ZUC-256-v2 with low time and data complexities, respectively. These attacks target the initialization phase and work in the related-key model with weak keys. Moreover, with a novel IV-correcting technique, we show how to efficiently recover at least 16 key bits for 15-round ZUC-256 and 14-round ZUC-256-v2 in the related-key setting, respectively. It is unpredictable whether our attacks can be further extended to more rounds with more advanced techniques. Based on the current attacks, we believe that the full 33 initialization rounds provide marginal security.

Elliptic Curve Hidden Number Problem (EC-HNP) was first introduced by Boneh, Halevi and Howgrave-Graham at Asiacrypt 2001. To rigorously assess the bit security of the Diffie--Hellman key exchange with elliptic curves (ECDH), the Diffie--Hellman variant of EC-HNP, regarded as an elliptic curve analogy of the Hidden Number Problem (HNP), was presented at PKC 2017. This variant can also be used for practical cryptanalysis of ECDH key exchange in the situation of side-channel attacks. In this paper, we revisit the Coppersmith method for solving the involved modular multivariate polynomials in the Diffie--Hellman variant of EC-HNP and demonstrate that, for any given positive integer $d$, a given sufficiently large prime $p$, and a fixed elliptic curve over the prime field $\mathbb{F}_p$, if there is an oracle that outputs about $\frac{1}{d+1}$ of the most (least) significant bits of the $x$-coordinate of the ECDH key, then one can give a heuristic algorithm to compute all the bits within polynomial time in $\log_2 p$. When $d>1$, the heuristic result $\frac{1}{d+1}$ significantly outperforms both the rigorous bound $\frac{5}{6}$ and heuristic bound $\frac{1}{2}$. Due to the heuristics involved in the Coppersmith method, we do not get the ECDH bit security on a fixed curve. However, we experimentally verify the effectiveness of the heuristics on NIST curves for small dimension lattices.

By exploiting the feature of partial nonlinear layers, we propose a new technique called algebraic meet-in-the-middle (MITM) attack to analyze the security of LowMC, which can reduce the memory complexity of the simple difference enumeration attack over the state-of-the-art. Moreover, while an efficient algebraic technique to retrieve the full key from a differential trail of LowMC has been proposed at CRYPTO 2021, its time complexity is still exponential in the key size. In this work, we show how to reduce it to constant time when there are a sufficiently large number of active S-boxes in the trail. With the above new techniques, the attacks on LowMC and LowMC-M published at CRYPTO 2021 are further improved, and some LowMC instances could be broken for the first time. Our results seem to indicate that partial nonlinear layers are still not well-understood.

Rasta and Dasta are two fully homomorphic encryption friendly symmetric-key primitives proposed at CRYPTO 2018 and ToSC 2020, respectively. It can be found from the designers’ analysis that the security of the two ciphers highly relies on the high algebraic degree of the inverse of the n -bit $$\chi $$ χ operation denoted by $$\chi _n^{-1}$$ χ n - 1 , while surprisingly the explicit formula of $$\chi _n^{-1}$$ χ n - 1 has never been given in the literature. As the first contribution, for the first time, we give a very simple formula of $$\chi _n^{-1}$$ χ n - 1 that can be written down in only one line and we prove its correctness in a rigorous way. Based on this formula of $$\chi _n^{-1}$$ χ n - 1 , an obvious yet important weakness of the two ciphers can be identified, which shows that their security against the algebraic attack cannot be solely based on the high degree of $$\chi _n^{-1}$$ χ n - 1 . Specifically, this weakness enables us to theoretically break two out of three instances of full Agrasta, which is the aggressive version of Rasta with the block size only slightly larger than the security level in bits. We further reveal that Dasta is more vulnerable against our attacks than Rasta because of its usage of a linear layer composed of an ever-changing bit permutation and a deterministic linear transform. Based on our cryptanalysis, the security margins of Dasta and Rasta parameterized with $$(n,\kappa ,r)\in \{(327,80,4),(1877,128,4),(3545,256,5)\}$$ ( n , κ , r ) ∈ { ( 327 , 80 , 4 ) , ( 1877 , 128 , 4 ) , ( 3545 , 256 , 5 ) } are reduced to only 1 round, where n , $$\kappa $$ κ and r denote the block size, the claimed security level and the number of rounds, respectively. These parameters are of particular interest as the corresponding ANDdepth is the lowest among those that can be implemented in reasonable time and target the same claimed security level.

It has been common knowledge that for a stream cipher to be secure against generic TMD tradeoff attacks, the size of its internal state in bits needs to be at least twice the size of the length of its secret key. In FSE 2015, Armknecht and Mikhalev however proposed the stream cipher Sprout with a Grain-like architecture, whose internal state was equal in size with its secret key and yet resistant against TMD attacks. Although Sprout had other weaknesses, it germinated a sequence of stream cipher designs like Lizard and Plantlet with short internal states. Both these designs have had cryptanalytic results reported against them. In this paper, we propose the stream cipher Atom that has an internal state of 159 bits and offers a security of 128 bits. Atom uses two key filters simultaneously to thwart certain cryptanalytic attacks that have been recently reported against keystream generators. In addition, we found that our design is one of the smallest stream ciphers that offers this security level, and we prove in this paper that Atom resists all the attacks that have been proposed against stream ciphers so far in literature. On the face of it, Atom also builds on the basic structure of the Grain family of stream ciphers. However, we try to prove that by including the additional key filter in the architecture of Atom we can make it immune to all cryptanalytic advances proposed against stream ciphers in recent cryptographic literature.

Rasta and Dasta are two fully homomorphic encryption friendly symmetric-key primitives proposed at CRYPTO 2018 and ToSC 2020, respectively. We point out that the designers of Rasta and Dasta neglected an important property of the $\chi$ operation. Combined with the special structure of Rasta and Dasta, this property directly leads to significantly improved algebraic cryptanalysis. Especially, it enables us to theoretically break 2 out of 3 instances of full Agrasta, which is the aggressive version of Rasta with the block size only slightly larger than the security level in bits. We further reveal that Dasta is more vulnerable against our attacks than Rasta for its usage of a linear layer composed of an ever-changing bit permutation and a deterministic linear transform. Based on our cryptanalysis, the security margins of Dasta and Rasta parameterized with $(n,\kappa,r)\in\{(327,80,4),(1877,128,4),(3545,256,5)\}$ are reduced to only 1 round, where $n$, $\kappa$ and $r$ denote the block size, the claimed security level and the number of rounds, respectively. These parameters are of particular interest as the corresponding ANDdepth is the lowest among those that can be implemented in reasonable time and target the same claimed security level.

Let $(N,e)$ be an RSA public key, where $N=pq$ is the product of equal bitsize primes $p,q$. Let $d_p, d_q$ be the corresponding secret CRT-RSA exponents. Using a Coppersmith-type attack, Takayasu, Lu and Peng (TLP) recently showed that one obtains the factorization of $N$ in polynomial time, provided that $d_p, d_q \leq N^{0.122}$. Building on the TLP attack, we show the first {\em Partial Key Exposure} attack on short secret exponent CRT-RSA. Namely, let $N^{0.122} \leq d_p, d_q \leq N^{0.5}$. Then we show that a constant known fraction of the least significant bits (LSBs) of both $d_p, d_q$ suffices to factor $N$ in polynomial time. Naturally, the larger $d_p,d_q$, the more LSBs are required. E.g. if $d_p, d_q$ are of size $N^{0.13}$, then we have to know roughly a $\frac 1 5$-fraction of their LSBs, whereas for $d_p, d_q$ of size $N^{0.2}$ we require already knowledge of a $\frac 2 3$-LSB fraction. Eventually, if $d_p, d_q$ are of full size $N^{0.5}$, we have to know all of their bits. Notice that as a side-product of our result we obtain a heuristic deterministic polynomial time factorization algorithm on input $(N,e,d_p,d_q)$.

At ToSC 2021, Rohit et al. presented the first distinguishing and key recovery attacks on 7 rounds Ascon without violating the designer’s security claims of nonce-respecting setting and data limit of 264 blocks per key. So far, these are the best attacks on 7 rounds Ascon. However, the distinguishers require (impractical) 260 data while the data complexity of key recovery attacks exactly equals 264. Whether there are any practical distinguishers and key recovery attacks (with data less than 264) on 7 rounds Ascon is still an open problem.In this work, we give positive answers to these questions by providing a comprehensive security analysis of Ascon in the weak key setting. Our first major result is the 7-round cube distinguishers with complexities 246 and 233 which work for 282 and 263 keys, respectively. Notably, we show that such weak keys exist for any choice (out of 64) of 46 and 33 specifically chosen nonce variables. In addition, we improve the data complexities of existing distinguishers for 5, 6 and 7 rounds by a factor of 28, 216 and 227, respectively. Our second contribution is a new theoretical framework for weak keys of Ascon which is solely based on the algebraic degree. Based on our construction, we identify 2127.99, 2127.97 and 2116.34 weak keys (out of 2128) for 5, 6 and 7 rounds, respectively. Next, we present two key recovery attacks on 7 rounds with different attack complexities. The best attack can recover the secret key with 263 data, 269 bits of memory and 2115.2 time. Our attacks are far from threatening the security of full 12 rounds Ascon, but we expect that they provide new insights into Ascon’s security.

The Modular Inversion Hidden Number Problem (MIHNP), introduced by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001, is briefly described as follows: Let $${\mathrm {MSB}}_{\delta }(z)$$ refer to the $$\delta $$ most significant bits of z. Given many samples $$\left( t_{i}, {\mathrm {MSB}}_{\delta }((\alpha + t_{i})^{-1} \bmod {p})\right) $$ for random $$t_i \in \mathbb {Z}_p$$, the goal is to recover the hidden number $$\alpha \in \mathbb {Z}_p$$. MIHNP is an important class of Hidden Number Problem.In this paper, we revisit the Coppersmith technique for solving a class of modular polynomial equations, which is respectively derived from the recovering problem of the hidden number $$\alpha $$ in MIHNP. For any positive integer constant d, let integer $$n=d^{3+o(1)}$$. Given a sufficiently large modulus p, $$n+1$$ samples of MIHNP, we present a heuristic algorithm to recover the hidden number $$\alpha $$ with a probability close to 1 when $$\delta /\log _2 p>\frac{1}{d\,+\,1}+o(\frac{1}{d})$$. The overall time complexity of attack is polynomial in $$\log _2 p$$, where the complexity of the LLL algorithm grows as $$d^{\mathcal {O}(d)}$$ and the complexity of the Gröbner basis computation grows as $$(2d)^{\mathcal {O}(n^2)}$$. When $$d> 2$$, this asymptotic bound outperforms $$\delta /\log _2 p>\frac{1}{3}$$ which is the asymptotic bound proposed by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001. It is the first time that a better bound for solving MIHNP is given, which implies that the conjecture that MIHNP is hard whenever $$\delta /\log _2 p<\frac{1}{3}$$ is broken. Moreover, we also get the best result for attacking the Inversive Congruential Generator (ICG) up to now.

MDS matrices are used in the design of diffusion layers in many block ciphers and hash functions due to their optimal branch number. But MDS matrices, in general, have costly implementations. So in search for efficiently implementable MDS matrices, there have been many proposals. In particular, circulant, Hadamard, and recursive MDS matrices from companion matrices have been widely studied. In a recent work, recursive MDS matrices from sparse DSI matrices are studied, which are of interest due to their low fixed cost in hardware implementation. In this paper, we present results on the exhaustive search for (recursive) MDS matrices over GL(4, F2). Specifically, circulant MDS matrices of order 4, 5, 6, 7, 8; Hadamard MDS matrices of order 4, 8; recursive MDS matrices from companion matrices of order 4; recursive MDS matrices from sparse DSI matrices of order 4, 5, 6, 7, 8 are considered. It is to be noted that the exhaustive search is impractical with a naive approach. We first use some linear algebra tools to restrict the search to a smaller domain and then apply some space-time trade-off techniques to get the solutions. From the set of solutions in the restricted domain, one can easily generate all the solutions in the full domain. From the experimental results, we can see the (non) existence of (involutory) MDS matrices for the choices mentioned above. In particular, over GL(4, F2), we provide companion matrices of order 4 that yield involutory MDS matrices, circulant MDS matrices of order 8, and establish the nonexistence of involutory circulant MDS matrices of order 6, 8, circulant MDS matrices of order 7, sparse DSI matrices of order 4 that yield involutory MDS matrices, and sparse DSI matrices of order 5, 6, 7, 8 that yield MDS matrices. To the best of our knowledge, these results were not known before. For the choices mentioned above, if such MDS matrices exist, we provide base sets of MDS matrices, from which all the MDS matrices with the least cost (with respect to d-XOR and s-XOR counts) can be obtained. We also take this opportunity to present some results on the search for sparse DSI matrices over finite fields that yield MDS matrices. We establish that there is no sparse DSI matrix S of order 8 over F28 such that S8 is MDS.